I would suggest that you charge them separately, especially if they are different types of batteries as is often the case. Even if they are the same type, the two banks will often be in different charge states and require different voltage and current regulation for optimal charging. Eg, the house may be deeply discharged and require a different voltage than the starter which is close to float levels.

I disagree. What is optimal for each battery separately is not optimal for the system consisting of both. Think of them as a single bank consisting of paralleled batteries. The charging current will divide between them according to their needs.